Question 722950
<ol><li>{{{sqrt(x-1)}}}<ul><li>Domain: "x" is in the radicand of an even-numbered root (square root is a "2nd root"). The radicand of an even-numbered root must not be negative. (IOW: It must be greater than or equal to zero.) So:
{{{x-1 >= 0}}}
or
{{{x >= 1}}}
This is the domain.</li><li>Range: The expression as a whole is a square root. This cannot be negative, either. So the range is zero and all positive numbers.</li></ul></li><li>{{{(10^x)+3}}}<ul><li>Domain: "x" is in an exponent. Exponents can be any number. So the domain is all real numbers.</li><li>Range. A power of 10 can never be zero or negative. (IOW: A power of 10 must be positive.) It can be <i>any</i> positive number. So {{{10^x > 0}}}. If we add three to each side we get: {{{10^x + 3 > 3}}}. On the left side we have the expression we started with. So this inequality tells us that the range is all numbers greater than 3.</li></ul></li><li>{{{log(2, (x-3))}}}<ul><li>"x" is in the argument of a logarithm. Arguments of logarithms must be positive. So
x - 3 > 0
or
x > 3
This is the domain.</li><li>Range: The expression as a whole is a logarithm. The value of a logarithm can be any real number so the range is all real numbers.</li></ul></li></ol>